Richards J.A., Jia X. Remote Sensing Digital Image Analysis: An Introduction

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5.9 Detecting Geometric Properties 131

5.9.2

Spatial Correlation – The Semivariogram

The semivariogram is a useful means for describing the spatial properties of an image

(or the scene being imaged) in a speciﬁed direction. It is constructed in the following

manner, by computing the average semivariance in the given direction according to

(Curran, 1988)



i=1

[φ(i) − φ(i + h)]

In this expression h represents the distance, or lag, between two pixels whose bright-

ness values, φ(i) and φ(i +h) are subtracted and then squared. By moving along the

given direction by pixel (i = 1, 2,...m) one half of the averaged squared distance

is computed, as indicated in the formula; m is the total number of pairs which are

separated by h. By varying the lag, h, a graph of the average semivariance versus lag

can be constructed as depicted in Fig. 5.13. That graph is called a semivariogram.

In a sense the semivariance measure is detecting how dissimilar pixel brightnesses

are, on the average, when separated by a lag h. If there is spatial periodicity in the

landscape then the semivariogram will reﬂect that behaviour as well.

The semivariogram for the image of Fig. 5.12a is shown in Fig. 5.13 for the four

regions chosen.

Several properties can be derived from the semivariogram; these are best illus-

trated by the idealised form shown in Fig. 5.14, and include the sill (its asymptotic

maximum value, if it exists), the nuggett variance (the extrapolated point of inter-

section with the ordinate), sometimes taken to indicate the noise properties of the

image since it represents variance that is not related to the spatial properties of the

scene, and the range, which is the lag or separation at which the sill is reached.

0.5

1.5

2.5

3.5

0 5 10 15 20

Semivariogram

Forest

Suburbs

Grass

Mountain

Fig. 5.13. Semivariogram for the image of Fig. 5.12a

132 5 Geometric Enhancement Using Image Domain Techniques

lag

semi-variance

nuggett variance

sill

range

Fig. 5.14. The idealised semivariogram for a region in which there is no spatial periodicity

5.9.3

Shape Detection

Recognition of shapes in image data has not been considered in remote sensing as

extensively as it has in object recognition exercises, such as robot vision. Presumably

this is because the resolution generally available in the past has been insufﬁcient to

deﬁne shape with any degree of precision. However with ground resolution elements

better than 20 to 30 m in imagery, shapes such as those of rectangular ﬁelds in

agriculture, and circular pivotal irrigation systems are quite well deﬁned.

Shape recognition can be carried out using template techniques, in which the

templates are chosen according to the shape of interest (Hord, 1982). The operation

required is one of correlation and not the convolution operation of (5.4). Correlation

is deﬁned by that same expression but with additions in place of subtractions. A

major difﬁculty with this approach, which as a consequence renders the technique

of limited value in practice, is that the template must match not only the shape of

interest, but also its size and orientation. Other methods therefore are often employed.

These include the adoption of shape factors (Underwood, 1970), moments of area

(Pavlidis, 1978) and Fourier transforms of shape boundaries (Pavlidis, 1980). In each

of these the shape must ﬁrst be delineated from the rest of the image. This is achieved

by edge and line detection processes.

References for Chapter 5

The template techniques that form the basis of much of the material presented in this chapter

are treated also by Castleman (1996), Moik (1980) and Hord (1982). Castleman also provides a

detailed linear systems theory approach to ﬁlter design. Gonzalez and Woods (1992) present the

method in a vector formulation, noting that the convolution operation in (5.1) can be expressed

References for Chapter 5 133

as the scalar product of the vector of template entries and the vector of pixel brightnesses

currently covered by the template. Such an approach allows templates to be designed to detect

combinations of lines and edges. Since the template entries are expressed in vector form they

can be used to deﬁne vector sub-spaces into which an image has projections. A large projection

into an edge sub-space implies edges in the image of the pixel currently being assessed, and

so on. This is assessed in terms of the vector angle between the image pixel vector and the

subspace basis vectors (template entries).

Gradient methods are covered also by Moik (1980), Gonzalez and Woods (1992), Hord

(1982) and to an extent by Castleman (1996). Gonzalez and Woods also include discussions

on the use of thresholds applied to the response of gradient operators. Vanderbrug (1976) and

Gurney (1980) consider the properties of nonlinear and semilinear line detecting templates.

Paine and Lodwick (1989) provide a good discussion of the application of edge detection

methods, while Brzakovic et al. (1991) consider the use of rule-based methods to assist in

edge detection when a number of templates is involved. While the edge detection methods

treated here have been applicable only to single bands of data, Cumani (1991) and Drewnick

(1994) have proposed operators for use on multispectral data.

K.R. Babu and R. Nevatia, 1980: Linear Feature Extraction and Description. Computer Graph-

ics and Image Processing, 13, 257–269.

E.O. Brigham, 1974: The Fast Fourier Transform, N.J. Prentice-Hall.

E.O. Brigham, 1988: The Fast Fourier Transform and its Applications, N.J. Prentice-Hall.

D. Brzakovic, R. Patton and R.L. Wang, 1991: Rule-based Multitemplate Edge Detector.

CVGIP: Graphical Models and Image Processing, 53, 258–268.

K.R. Castleman, 1996: Digital Image Processing, N.J. Prentice-Hall.

A. Cumani, 1991: Edge Detection in Multispectral Images. CVGIP: Computer Models and

Image Processing, 53, 40–51.

C. Drewnick, 1994: Multispectral Edge Detection. Some Experiments on Data from Landsat

TM. Int. J. Remote Sensing, 15, 3743–3765.

E.M. Eliason and A.S. McEwan, 1990: Adaptive Box Filters for Removal of Random Noise

from Digital Images. Photogrammetric Engineering and Remote Sensing, 56, 453–458.

R.C. Gonzalez and R.E. Woods, 1992: Digital Image Processing, Mass., Addison-Wesley.

C.M. Gurney, 1980: Threshold Selection for Line Detection AIgorithms. IEEE Trans. Geo-

science and Remote Sensing, GE-18, 204–211.

R.M. Haralick, 1979: Statistical and Structural Approaches to Texture. Proc. IEEE, 67, 786–

802.

R.M. Hord, 1982: Digital Image Processing of Remotely Sensed Data, N.Y. Academic.

C.D. McGillem and G.R. Cooper, 1984: Continuous and Discrete Signal and SystemsAnalysis,

2e, N.Y., Holt, Reinhard and Winston.

J.G. Moik, 1980: Digital Processing of Remotely Sensed Images, N.Y., Academic.

S.H. Paine and G.D. Lodwick, 1989: Edge Detection and Processing of Remotely Sensed

Digital Images. Photogrammetria (PRS), 43, 323–336.

C. Paul and K.S. Shanmugan, 1982: A Fast Thinning Operator. IEEE Trans. Systems, Man,

Cybernetics, SMC-12, 567–569.

T. Pavlidis, 1978: A Review of Algorithms for Shape Analysis. Computer Graphics and Image

Processing, 7, 243–258.

T. Pavlidis, 1980: Algorithms for Shape Analysis of Contours and Waveforms. IEEE Trans.

Pattern Analysis and Machine Intelligence, PAMI-2, 301–312.

A. Rosenfeld, 1978: Image Processing and Recognition, Technical Report 664, Computer

Vision Laboratory, University of Maryland.

134 5 Geometric Enhancement Using Image Domain Techniques

A. Rosenfeld and M. Thurston, 1971: Edge and Curve Detection for Visual Scene Analysis,

IEEE Trans. Computers, C-20, 562–569.

E.E. Underwood, 1970: Quantitative Stereology, Mass., Addison-Wesley.

G.J. Vanderbrug, 1976: Line Detection in Satellite Imagery, IEEE Trans. Geoscience Elec-

tronics, GE-14, 37–44.

Problems

5.1 The template entries for line and edge detection sum to zero whereas those for smoothing

do not. Why do you think that is so?

5.2 Repeat the example of Fig. 5.10 but by using a [5 × 1] smoothing operation in part (a),

rather than [3 × 1] smoothing.

5.3 Repeat the example of Fig. 5.10 but by using a [3 × 1] median ﬁltering operation in part

(a) rather than [3 × 1] mean value smoothing.

5.4 An alternative smoothing process to median and mean value ﬁltering using template

methods is known as modal ﬁltering. In this approach a pixel at the centre of the template

neighbourhood is replaced by the brightness value that occurs most frequently in the neigh-

bourhood. Apply [3×1]and [5×1]modal ﬁlters to the image data of Fig. 5.6. Note differences

in the results compared with mean value and median smoothing, particularly around the edges.

5.5 Suppose S is a template operation that implements smoothing and O is the template

operator that leaves an image unchanged (see Sect. 5.8). Then an edge enhanced image created

by the subtractive smoothing approach of Sect. 5.6.4 can be expressed according to

New image = O (old image) + O (old image) − S (old image)

Rewrite this expression to incorporate two user deﬁned parameters α and β that will cause the

formula to implement any of smoothing, edge detection or edge enhancement.

5.6 (Requires vector algebra background). Show that template methods for line and edge

detection can be expressed as the scalar product of a vector composed from the template entries

and a vector formed from the neighbourhood of pixels currently covered by the template. Show

how the angle between the template and pixel vectors can be used to assess the edge or line

feature a current pixel most closely corresponds to. (See Gonzalez and Woods (1992)).

5.7 The following kernel is sometimes convolved with image data. What operation will it

implement?

5.8 Consider the middle pixel shown in the ﬁgure below and calculate its new value if

(i) a 3 ×3 median ﬁltering is applied,

(ii) a 3 ×3 template which performs edge enhancement by subtractive smoothing is applied,

Problems 135

(iii) a 3 ×1 image smoothing template with a threshold 2 is applied,

(iv) the Sobel operator is applied for edge detection.

5.9 Image smoothing can be performed by template operators that implement averaging or

median ﬁltering. Compare the methods. particularly as they affect edges. Would you expect

median ﬁltering to be useful in edge enhancement by the technique of subtracting a smoothed

image from the original?

5.10 Ifa3× 3 smoothing template is applied to an image twice in succession, how many

neighbours will have played a part in modifying the brightness of a given pixel? Design a

single template to achieve the same result in one pass.

Multispectral Transformations

of Image Data

The multispectral or vector character of most remote sensing image data renders it

amenable to spectral transformations that generate new sets of image components

or bands. These components then represent an alternative description of the data, in

which the new components of a pixel vector are related to its old brightness values in

the original set of spectral bands via a linear operation. The transformed image may

make evident features not discernable in the original data or alternatively it might

be possible to preserve the essential information content of the image (for a given

application) with a reduced number of the transformed dimensions. The last point

has signiﬁcance for displaying data in the three dimensions available on a colour

monitor or in colour hardcopy, and for transmission and storage of data.

The role of this chapter is to present image transformations of value in the en-

hancement of remote sensing imagery, although some also ﬁnd application in pre-

conditioning image data prior to classiﬁcation by the techniques of Chaps. 8 and 9.

The techniques covered, which appeal directly to the vector nature of the image,

include the principal components transformation and so-called band arithmetic. The

latter includes the creation of ratio images. Some specialised transformations, such

as the Kauth-Thomas tasseled cap transform are also treated.

6.1

The Principal Components Transformation

The multispectral or multidimensional nature of remote sensing image data can be

accommodated by constructing a vector space with as many axes or dimensions as

there are spectral components associated with each pixel. In the case of Landsat

Thematic Mapper data it will have seven dimensions while for SPOT HRV data

it will be three dimensional. For hyperspectral data there may be several hundred

axes. A particular pixel in an image is plotted as a point in such a space with co-

ordinates that correspond to the brightness values of the pixels in the appropriate

spectral components. For simplicity the treatment to be developed in this topic will

138 6 Multispectral Transformations of Image Data

be based upon a two dimensional multispectral space (say visible red and infrared)

since the diagrams are then easily understood and the mathematical detail is readily

assimilated. The results derived however are perfectly general and apply to data of

any dimensionality.

6.1.1

The Mean Vector and Covariance Matrix

The positions of pixel points in multispectral space can be described by vectors,

whose components are the individual spectral responses in each band. Strictly, these

are vectors drawn from the origin to the pixel point as seen in Appendix D, but this

concept is not used explicitly. Consider a mu1tispectral space with a large number of

pixels plotted in it as shown in Fig. 6.1, with each pixel described by its appropriate

vector x. The mean position of the pixels in the space is deﬁned by the expected

value of the pixel vector x, according to

m = E(x}=



k=1

(6.1)

where m is the mean pixel vector and the x

are the individual pixel vectors of total

number K;E is the expectation operator.

While the mean vector is useful to deﬁne the average or expected position of the

pixels in multispectral vector space, it is of value to have available a means by which

their scatter or spread is described. This is the role of the covariance matrix which is

deﬁned as

= E{(x − m)(x − m)

} (6.2a)

in which the superscript ‘t’ denotes vector transpose. (See Appendix D).

An unbiased estimate of the covariance matrix is given by

K − 1



k=1

− m)(x

− m)

(6.2b)

The covariance matrix is one of the most important mathematical concepts in

the analysis of multispectral remote sensing data, as a result of which it is of value

Fig. 6.1. Two dimensional multispectral space show-

ing the individual pixel vectors and their mean posi-

tion, as deﬁned by m, the mean vector

6.1 The Principal Components Transformation 139

Fig. 6.2. Two dimensional data showing no correlation between components a and high

correlation between components b

to consider some sample calculations to enable its properties to be emphasised. In

these it will be seen that if there is correlation between the responses in a pair of

spectral bands the corresponding off-diagonal element in the covariance matrix will

be large by comparison to the diagonal terms. On the other hand, if there is a little

correlation, the off-diagonal terms will be close to zero. This behaviour can also be

described in terms of the correlation matrix R whose elements are related to those

of the covariance matrix by



= v

√

(6.3)

where 

is an element of the correlation matrix and v

etc. are elements of the

covariance matrix; v

and v

are the variances of the ith and jth bands of data. The



describe the correlation between band i and band j.

Consider the two, two-dimensional sets of data shown in Fig. 6.2. That in Fig. 6.2a

shows little correlation between the two components: in other words, both compo-

nents are necessary to describe where a pixel lies in the space. The data shown in

Fig. 6.2b however exhibits a high degree of correlation between its two components,

evident in the elongated spread of the data at an angle to the axes. One dimension on

its own is almost sufﬁcient to predict where a pixel lies in the space, and an increase

or decrease in either component suggests a corresponding increase or decrease in

the other. This is not the case with Fig. 6.2a. In terms of the individual images cor-

responding to the bands of multispectral data, highly correlated bands as depicted

in Fig. 6.2b would yield image components very similar in appearance. Where one

is dark the other will be dark and so on. The image components corresponding to

Fig. 6.2a, however, would display no similar consistently common behaviour. Prob-

lem 6.7 shows a number of other situations of high and low correlation. Importantly,

if the data is scattered in an elongated fashion, as seen in Fig. 6.2b, but the directions

of major scatter are parallel to the coordinate (measurement) axes, then there is little

correlation among the measurements.

Table 6.1 shows a sample set of hand calculations undertaken to ﬁnd the covari-

ance and correlation matrices for Fig. 6.2a. Normally this would be carried out by

computer, particularly for data with higher dimensionality. As noted from the corre-

140 6 Multispectral Transformations of Image Data

Table 6.1. Computation of covariance and correlation matrices for Fig. 6.2a

The mean vector is m =



3.00

2.33



lation matrix there is no correlation between the individual components of the data,

a fact which is evident also in the zero off-diagonal entries in the covariance matrix.

The entry of 2.40 in the upper left hand corner of the covariance matrix signiﬁes that

the data points have a variance of 2.40 along the horizontal axis, or a standard devia-

tion of 1.55 about the mean. Similarly, the variance and standard deviation vertically

are 1.87 and 1.37 respectively.

For the data in Fig. 6.2b, it is shown by a similar set of calculations to those in

Table 6.1 that

m =



3.50





1.900 1.100

1.100 1.100



and

R =



1.000 0.761

0.761 1.000



Thus components 1 and 2 of the data in Fig. 6.2b are 76% correlated.

It should be noted that both the covariance and correlation matrices are symmet-

ric and that an image data set, in which there is no correlation between any of its

multispectral components, will have a diagonal covariance (and correlation) matrix.

6.1 The Principal Components Transformation 141

6.1.2

A Zero Correlation, Rotational Transform

It is fundamental to the development of the principal components transformation to

ask whether there is a new co-ordinate system in the multispectral vector space in

which the data can be represented without correlation; in other words, such that the

covariance matrix in the new co-ordinate system is diagonal. For a particular two

dimensional vector space such a new co-ordinate system is depicted in Fig. 6.3. If

the vectors describing the pixel points are represented as y in the new co-ordinate

system then it is desired to ﬁnd a linear transformation G of the original co-ordinates,

such that

y = Gx = D

x (6.4)

subject to the constraint that the covariance matrix of the pixel data in y space is

diagonal. Expressing G as D

will make the comparison of principal components

with other transformation operations, treated later, much simpler.

In y space the covariance matrix is, by deﬁnition,

= E{(y − m

)(y − m

)

}

where m

is the mean vector expressed in terms of the y co-ordinates. It is shown

readily that

= E{y}=E{D

x}=D

E{x}=D

Fig. 6.3. Illustration of a modiﬁed co-ordinate system in which the pixel vectors have uncor-

related components

E{D

x}=



k=1

= D



k=1

= D

i.e. D

, being a matrix of constants, can be taken outside an expectation operator.